Hyperbolicity cones are amenable

TL;DR

This paper proves that all hyperbolicity cones possess the property of amenability, extending known results from spectrahedral cones and revealing structural properties of hyperbolicity cones.

Contribution

The paper establishes that every hyperbolicity cone is amenable and shows that faces and intersections of hyperbolicity cones are also hyperbolicity cones.

Findings

All hyperbolicity cones are amenable.

Faces of hyperbolicity cones are hyperbolicity cones.

Intersections of hyperbolicity cones are hyperbolicity cones.

Abstract

Amenability is a notion of facial exposedness for convex cones that is stronger than being facially dual complete (or "nice") which is, in turn, stronger than merely being facially exposed. Hyperbolicity cones are a family of algebraically structured closed convex cones that contain all spectrahedral cones (linear sections of positive semidefinite cones) as special cases. It is known that all spectrahedral cones are amenable. We establish that all hyperbolicity cones are amenable. As part of the argument, we show that any face of a hyperbolicity cone is a hyperbolicity cone. As a corollary, we show that the intersection of two hyperbolicity cones, not necessarily sharing a common relative interior point, is a hyperbolicity cone.